A346979 Count of the prime decimal descendants of n.

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset

0,1

Comments

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.

a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).

a(A032352(m)) = 0; a(A119289(m)) = 0.

A214342 is a subset, as A214342(m) = a(prime(m)).

Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

Links

Table of n, a(n) for n=0..76.

Example

a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339

Mathematica

Table[Length@Rest@Flatten[FixedPointList[(b=#; Select[Flatten[(a=#; FromDigits/@(Join[IntegerDigits@a, {#}]&/@If[b=={0}, Range@9, {1, 3, 7, 9}]))&/@b], PrimeQ])&, {n}]], {n, 0, 76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

Prog

(Python)
from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))

Crossrefs

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Sequence in context: A317915 A033403 A174092 * A134456 A247442 A106428

Adjacent sequences: A346976 A346977 A346978 * A346980 A346981 A346982

Keyword

nonn,base

Author

Ya-Ping Lu, Aug 09 2021

Status

approved